What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2, in the form ax² + bx + c = 0, where a ≠ 0. The solutions (roots) represent the x-intercepts of the parabola y = ax² + bx + c.
The Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a. This formula always works for any quadratic equation. Enter a, b, and c — the formula gives both roots directly. The term b² - 4ac is called the discriminant (Δ).
Understanding the Discriminant
Δ > 0: Two distinct real roots (parabola crosses x-axis at two points). Δ = 0: One repeated real root (parabola is tangent to x-axis). Δ < 0: Two complex conjugate roots (parabola does not cross x-axis). Our calculator handles all three cases, displaying complex roots as a ± bi.
Real-World Applications
Quadratic equations appear everywhere: projectile motion (when does a ball land?), optimization (maximizing profit/area), circuit analysis (resonant frequency), and computer graphics (ray-sphere intersection). Understanding them is fundamental to physics, engineering, and machine learning.